Uniform Convergence: Does Not Converge on (0,1)

[hth]

Homework Statement

Let f_n(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly.

Homework Equations

The Attempt at a Solution

I know why it is not uniformly convergent. Even though f_n(x) goes to zero monotonically on the interval (0,1), it's not continuous on a compact interval. How would go about showing this formally/by example?

 

[snipez90]

A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?

 

[hth]

A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?

Isn't that the solution to it if f_n(x) = x^(n)? How does that apply here?